Persistent Homology to Quantify the Quality of Surface‐Supported Covalent Networks

Order-Disorder Transition of Two-Dimensional Molecular Networks through a Stoichiometric Design

Materials with disordered structures may exhibit interesting properties. Metal-organic frameworks (MOFs) are a class of hybrid materials composed of metal nodes and coordinating organic linkers. Recently, there has been growing interest in MOFs with structural disorder and the investigations of amorphous structures on surfaces. Herein, we demonstrate a bottom-up method to construct disordered molecular networks on metal surfaces by selecting two organic molecule linkers with the same symmetry but different sizes for preparing two-component samples with different stoichiometric ratios. The amorphous networks are directly imaged by scanning tunneling microscopy under ultrahigh vacuum with a submolecular resolution, allowing us to quantify its degree of disorder and other structural properties. Furthermore, we resort to molecular dynamics simulations to understand the formation of the amorphous metal-organic networks. The results may advance our understanding of the mechanism of formation of monolayer molecular networks with structural disorders, facilitating the design and exploration of amorphous MOF materials with intriguing properties.

Persistent homology in two-dimensional atomic networks

The topology of two-dimensional network materials is investigated by persistent homology analysis. The constraint of two dimensions allows for a direct comparison of key persistent homology metrics (persistence diagrams, cycles, and Betti numbers) with more traditional metrics such as the ring-size distributions. Two different types of networks are employed in which the topology is manipulated systematically. In the first, comparatively rigid networks are generated for a triangle-raft model, which are representative of materials such as silica bilayers. In the second, more flexible networks are generated using a bond-switching algorithm, which are representative of materials such as graphene. Bands are identified in the persistence diagrams by reference to the length scales associated with distorted polygons. The triangle-raft models with the largest ordering allow specific bands B_n (n = 1, 2, 3, …) to be allocated to configurations of atoms separated by n bonds. The persistence diagrams for the more disordered network models also display bands albeit less pronounced. The persistent homology method thereby provides information on n-body correlations that is not accessible from structure factors or radial distribution functions. An analysis of the persistent cycles gives the primitive ring statistics, provided the level of disorder is not too large. The method also gives information on the regularity of rings that is unavailable from a ring-statistics analysis. The utility of the persistent homology method is demonstrated by its application to experimentally-obtained configurations of silica bilayers and graphene.